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Robust Connectivity of Graphs on Surfaces

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Extended Abstracts EuroComb 2021

Part of the book series: Trends in Mathematics ((RPCRMB,volume 14))

Abstract

Let \(\varLambda (T)\) denote the set of leaves in a tree T. One natural problem is to look for a spanning tree T of a given graph G such that \(\varLambda (T)\) is as large as possible. Recently, a similar but stronger notion called the robust connectivity of a graph G was introduced, which is defined as the minimum value \(\frac{|R \cap \varLambda (T)|}{|R|}\) taken over all nonempty subsets \({R\subseteq V(G)}\), where \(T = T(R)\) is a spanning tree on G chosen to maximize \({|R \cap \varLambda (T)|}\). We prove a tight asymptotic bound of \(\varOmega (\gamma ^{-\frac{1}{r}})\) for the robust connectivity of r-connected graphs of Euler genus \(\gamma \). Moreover, we give a surprising connection between the robust connectivity of graphs with an edge-maximal embedding in a surface and the surface connectivity of that surface, which describes to what extent large induced subgraphs of embedded graphs can be cut out from the surface without splitting the surface into multiple parts. For planar graphs, this connection provides an equivalent formulation of a long-standing conjecture of Albertson and Berman.

T.M. was supported by a postdoctoral fellowship at the Simon Fraser University through NSERC grants R611450 and R611368. J.N. received support for foreign internships for the WDSMCS students in the framework of “Excellence initiative – research university” programme at MIM, University of Warsaw. L.S. was supported by NSERC grant R611368. The full version of the paper can be found on arXiv [4].

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Notes

  1. 1.

    This parameter was formerly called “game connectivity” in [5]. However, we believe that the term “robust connectivity” better suits the properties of this parameter.

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Correspondence to Tomáš Masařík .

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Bradshaw, P., Masařík, T., Novotná, J., Stacho, L. (2021). Robust Connectivity of Graphs on Surfaces. In: Nešetřil, J., Perarnau, G., Rué, J., Serra, O. (eds) Extended Abstracts EuroComb 2021. Trends in Mathematics(), vol 14. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-83823-2_135

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